Problem 44 - Calculus Volume 1

Question

Use the limit definition of derivative to show that the derivative does not exist at \(x=a\) for each of the given functions. $$ f(x)=\frac{|x|}{x}, x=0 $$

Step-by-Step Solution

Answer

The derivative does not exist at \(x=0\) as the function is undefined there and the left and right limits differ.

Step 1: Define the Derivative Using the Limit Definition

The derivative of a function at a point is defined as the limit \( \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \). In this case, we want to find whether the derivative of \( f(x) = \frac{|x|}{x} \) exists at \( x = 0 \).

Step 2: Evaluate the Function at the Point of Interest

The function is \( f(x) = \frac{|x|}{x} \). Therefore, at \( x = 0 \), we have \( f(x) = \frac{|0|}{0} \), which is undefined. However, we can still use the limit to evaluate the derivative.

Step 3: Check the Limit from the Positive Side

Consider \( h > 0 \), so \( f(h) = \frac{|h|}{h} = \frac{h}{h} = 1 \). The limit as \( h \to 0^+ \) is:\[ \lim_{h \to 0^+} \frac{f(0+h) - f(0)}{h} = \lim_{h \to 0^+} \frac{1 - f(0)}{h} = \lim_{h \to 0^+} \frac{1 - \text{undefined}}{h} \]

Step 4: Check the Limit from the Negative Side

Consider \( h < 0 \), so \( f(h) = \frac{|h|}{h} = \frac{-h}{h} = -1 \). The limit as \( h \to 0^- \) is:\[ \lim_{h \to 0^-} \frac{f(0+h) - f(0)}{h} = \lim_{h \to 0^-} \frac{-1 - f(0)}{h} = \lim_{h \to 0^-} \frac{-1 - \text{undefined}}{h} \]

Step 5: Analyze and Conclude

Since \( f(0) \) is undefined, we cannot correctly compute \( \lim_{h \to 0} \frac{f(0+h) - f(0)}{h} \). Additionally, \( \lim_{h \to 0^+} \) results in \( \frac{1}{h} \) and \( \lim_{h \to 0^-} \) results in \( \frac{-1}{h} \), indicating different left-hand and right-hand limits, further confirming that the limit and thus the derivative does not exist.

Key Concepts

Derivative Existence

For a derivative to exist at a certain point, the left-hand limit and the right-hand limit of the derivative at that point must be equal. The limit definition of derivative is expressed as \( \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \). In simpler terms, this means that as we approach the point from both the left and the right, the slope of the tangent at that point should be the same.

For piecewise functions like \( f(x) = \frac{|x|}{x} \), determining the slope might not always be straightforward. This is especially true at critical points such as \( x=0 \) where the function behavior changes. We must investigate both one-sided limits, from the positive direction (right-hand limit) and the negative direction (left-hand limit), to see if they match.

If they differ, the derivative at that point does not exist. This is because the sudden change in direction means there is no unique slope at that point.

Right-Hand Limit

The right-hand limit looks at the behavior of the function as we approach a point from the positive side, namely from values greater than the point in question.

When calculating the derivative of \( f(x) = \frac{|x|}{x} \) at \( x = 0 \), we consider \( h > 0 \).

Here, when \( h \to 0^+ \), the function \( f(h) = \frac{|h|}{h} = 1 \). Thus, the derivative calculation becomes:

• \( \lim_{h \to 0^+} \frac{f(0+h) - f(0)}{h} \)
• As \( f(0) \) is undefined, this translates to \( \lim_{h \to 0^+} \frac{1}{h} \), a value that approaches infinity.

Such behavior where the limit approaches infinity indicates a vertical tangent, further supporting that the derivative at this point is non-existent.

Left-Hand Limit

The left-hand limit explores the function's behavior as we approach a point from the negative side, which means from values less than that point.

In the context of \( f(x) = \frac{|x|}{x} \) at \( x = 0 \), we focus on \( h < 0 \).

As \( h \to 0^- \), the function \( f(h) = \frac{-h}{h} = -1 \). The limit calculation becomes:

• \( \lim_{h \to 0^-} \frac{f(0+h) - f(0)}{h} \)
• Again, with \( f(0) \) undefined, this resolves to \( \lim_{h \to 0^-} \frac{-1}{h} \), representing negative infinity.

As with the right-hand limit, reaching negative infinity indicates a sharp turn at the point in question, reinforcing the conclusion that a derivative cannot be defined at \( x=0 \).

Piecewise Functions

Piecewise functions are defined by different expressions based on different intervals in their domain.

The function \( f(x) = \frac{|x|}{x} \) is a perfect example. It behaves differently depending on whether \( x \) is positive or negative.

• For \( x > 0 \), \( f(x) = 1 \).
• For \( x < 0 \), \( f(x) = -1 \).

At \( x = 0 \), the function is undefined. This undefined nature at certain critical points like \( x=0 \) makes it challenging to establish a uniform slope across the point.

Examining piecewise functions around points of discontinuity requires analyzing both left-hand and right-hand limits, just as we did for confirming the lack of a derivative at \( x=0 \). Understanding how piecewise functions alter at different intervals is crucial when determining derivative existence at points of interest.